Close to the old city, and looking for thai style and maybe with a swimming pool (for 2 teens)? Thanks
A dozen of good hotels in Chiang Mai old city where are matched your requirements.
I recommend 99 Gallery, Rachamankha and De Lanna Hotel.
Many of my friends using airbnb.com, give it a shot
I do recommend Top Garden Boutique Guesthouse. It's not close to the old town but it's a perfect option to get around for backpackers. You can view my itinerary for more info here.